Systems composed of other objects besides stars are currently ill-defined. For example, Pluto-Charon should technically be a binary dwarf planet system, but it is yet to be recognized as such by the IAU.
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called2voyage♦Dec 11 '13 at 16:53

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I have problem understanding what the question is. Status as defined by whom and for what purpose?
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TildalWaveDec 11 '13 at 18:09

Since Jupiter-Sun's center of mass lies outside the volume of the Sun, that means that the Sun moves around that center of mass All objects orbit at the barycenter between the two objects, not just Jupiter.
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asawyerDec 11 '13 at 18:43

3 Answers
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I'm not sure I understand your question entirely, but i'll do my best to offer a decent answer. It's true that the composition of Jupiter is very similar to that of the Sun (very similar approx. $H$ and $He$ abundance and pretty similar in density). The problem is that Jupiter is not nearly massive enough to have the internal pressure and temperature to undergo nuclear fusion. Jupiter doesn't have any particular special status aside from being the King of Planets in our solar system.

As for the last part of your question, all objects orbit around a center of mass. Though because the Sun is much more massive, the center of mass lies very close to the center of the star (except in the case of Jupiter, where the CoM lies outside of the sun and is approx the length of its radius). This is why all planets in our solar system orbit around the sun. This will indeed cause a slight perturbation of orbital alignment, but I don't believe it's significant.

You're right Jupiter is indeed the only exception (i'll go ahead and edit that in the answer just to be clear). Though the Sun is not a planet, so i'm not certain the double planet classification holds. Jupiter is also not a star from the reasons stated above so it's not a binary star system. I don't believe a special status name exists for this type of phenonmenon.
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Lame-Ov2.0Dec 11 '13 at 21:17